I have been puzzling over this problem for about a week now and cannot find the answer. In my opinion it is very theoretical, but I know I ma not the best mathematician on here so maybe someone else could look at this.

You need to derive the probability distribution for [tex]\bar{X}[/tex]. This is possible, since it is a sum of n quantities for which you know the distribution. Tthe distribution for [tex]\bar{X}[/tex] willl depend on n. You can then calculate the probability that [tex]|\bar{X}-\mu|>\sigma/2[/tex] in the ordinary way. This gives you an inequality for n. Then find the smallest n that satisfies the inequality.

When you know n, you have completely specified the distribution for [tex]\bar{X}[/tex], and you can do part (b).

So the first step is to find the distribution of a sum of quantities, expressed in terms of their individual distributions.

Cool, that's the same thing I got. From that you can write the explicit gaussian probability density for e.g. [tex]u:=\bar{X}-\mu[/tex]. Call it e.g. [tex]\rho_n(u)[/tex]. It will depend on n of course. It will simply the usual normalied gaussian centered around u=0, with variance [tex]\sigma^2/n[/tex] I think.

Then the probability [tex]P(n)[/tex] that [tex]|u|>\sigma/2[/tex] is given as a integral that you can solve using the error function: